Question

Consider the following linear transformation T: RS → R3 where T(X1, X2, X3, X4, X5) = (x1-X3+X4, 2X1+X2-X3+2x4, -2X1+3x3-3x4+x5) (a) Determine the standard matrix representation A of T(x).

Answer 1

Solution :- Consider the linear transpoemation 3 Tips ir where T (21,22,23,4,5) = (x,-23+24, 2+X2-23+2324 -2x, +373–324 +265 ) a) To find the standard matriz representation A Of TCX) for, let be s={(1,0,0,0,0), (0,1,0,0,0),(0,0,1,0,0), (0,0,0,1,0), (0,0,0,0, 1) } standard basis for IRS f s'= {(1,0,0), (0, 1, o), 60,0,1)} be standard basis for 1R3 a The of standard mateix representation A T(X) is by. given A={[7(e)]s! [TC@z)] s', [tle)]s!, [Teenis', [les) Is ] Tlei) = T(1,0,0,0,0) = (1-0+0,201) +0-0+0, -2(1) +0-0 +0) .. Tle) = (1, 2,-2) Tle) = (1,0,0) +2(0,1,0)-260,0,1)

... [TC)]şı = (1, 2, -2) T. The - Toil, o, o, o). -0,10) Cby deth of T). 5. Tlez) = O(1,0,0)+1(0,1,0) to (0,0,1). :( Tlex) 351 scoily o) T TE) - T(0,0,1,0,0) =(-1, 3) (by defn oq T. TC) = -1(1,0,0)-1(0,1,0) +3(0,0,1) :. [Teenis: - (-1,-1, 3) T -- Tlen) = T(0,0,0,1,0) = (1,2,3) Tlen) = |(1,0,0) +2(0,1,0) -3(0,0,1) > T .. [T(en)] si c (1,2,-3) T(es) = T(0,0,0,0, 12 -(0,0,1) .. Tles) = OCR, 0,0) +0(0,1,0) +IC0,0,1) Т. 6 :. [tles) Is = (0,0,1). from equation through ③ to ④ Equation 0 becomes.

- A= 2 - -) 2 -2 0 3 which is required standard matrix sepresentation of T(&).